The important questions in Statistics
1. A study of the English section reviews the following
contingency table:
Year I II III IV Total
Drink
Tea 200 500 300 200 1200
Coffee 100 400 500 100 1100
7 Up 300 400 400 100 1200
Total 600 1300 1200 400 3500
A student is selected randomly, what is the probability that:
1) He likes tea or 7-up.
1200 1200 0 2400
p(tea U 7-up)= p(tea)+p(7-up)-p(tea ∩ 7-up)=
3500 3500 3500 3500
2) He likes 7-up and he is a 4th year student
p(7-up ∩ 4th )= 100
3500
st
3) He is either a 1 year or he likes tea
600 1200 200 1600
p(1st U tea)=p(1st)+p(tea)-(p(1st ∩ tea)=
3500 3500 3500 3500
4) he is a 2nd year student and he likes Coffee
p(2nd ∩coffee)= 400
3500
rd
5) He is not a 3 year student.
p(3rd)= 1200 p(3rd) 1
1200 2300
3500 3500 3500
rd
6) He likes tea and he is a 3 year student
p(tea ∩ 3rd )= 300
3500
7) He likes tea given he is 2nd year student.
500
nd p(tea 2nd 3500 500
p(tea | 2 )=
p (2nd ) 1300 1300
3500
8) He is a 1st year or 3rd year student.
600 1200 1800
(1st U 3rd)=p(1st)+p(3rd )-(p(1st ∩ 3rd )=
3500 3500 3500
2. A box contains 5 black balls and 3 white balls, if we select
successively two balls without replacement
graph the probability tree of this experiment, and then find the
following probability:
1) The prob. of getting the two balls of the same color.
2) The prob. of that the second ball is black
3) The prob.of getting at least one white ball.
4) The prob. of getting the two balls of different color
4
7 B
4B
5 B 3W
8 3 W
5B 7
3W 5
5B 7 B
3 W
8 2W
2 W
7
1) (1st B ∩ 2nd B) U p(1st W ∩ 2nd W)
=
5 4 3 2 20 6 26 13
8 7 8 7 56 56 56 28
2) (1st B ∩ 2nd B) U p(1st W ∩ 2nd B)
= 5 8 4 7 38 5 7 56 15 56 8
20
56
35 5
3) (1st W ∩ 2nd B) + p(1st B ∩ 2nd W) + p(1st W ∩ 2nd W)
= 38 5 7 5 8 3 7 38 2 7
15 15 6 36 9
=
56 56 56 56 14
4) (1st W ∩ 2nd B) U p(1st B ∩ 2nd W)
= 38 5 7 5 8 3 7 15 15 30 15
56 56 56 28
3. Assume that P (A) = 0.2, P (B) = 0.6 and P (A∩B) = 0.12
Find the following probabilities:
1) P(AUB)
2) P(A-B)
3) P(A|B)
4) P ( A B)
5) Can you conclude that A and B are dependent or independent events?
6) Can you conclude that A and B are mutually exclusive or not?
1) P(AUB)=P(A)+P(B)- P(A∩B)
=0.2 + 0.6 – 0.12 = 0.68
2) P(A-B)=P(A)- P(A∩B)
=0.2 – 0.12 = 0.08
3) P(A|B)= P(A B 0.12 0.2
P ( B) 0.6
4) P ( A B) =1- P(AUB)
=1-0.68=0.32
5) P(A) X P(B) = 0.2 X 0.6=0.12
∴ P(A∩B)= P(A) X P(B)
∴ A and B are independent events
6) P(A∩B) ≠ zero
∴ A and B are not mutually exclusive
4. We have two boxes. The first contains two red balls and
three white balls, and the second box contains one red ball and
seven white balls.
If we randomly select a box and one ball was chosen from this box, and
we found that it was a white balls. What is the probability that this ball
was selected from the second box?
2
5 R
1 2R
2 B1 3W 3 W
5
5
1
1R 8 R
1 B2
2 7W
7 W
8 8
1 7 7
P(B2 W 2 8 16 35
P (B2|W) =
P (W ) 1 3 1 7 3 7 59
2 5 2 8 10 16
-from this example. But we found that it was a red ball;
What is the probability that ball was selected from the first box
1 2
P(B1 R 2 5 16
P (B1|R) =
P ( R) 1 2 1 1 21
2 5 2 8
5.Assume that 60% of the students of English section in future
academy live in heliopolis and Nasr City among those who
leave in helioplis and nasr city 70% likes video games among
the students who don't live there in heliopolis 80% likes vedio
games.
A student is selected randomly from all students of English section and
we were found that he likes video games. What is the probability that
he doesn't live in Heliopolis?
70% Like video games
60% Helioplis &Nacr City
30% Don't Like
40% 80% Like video games
Other Places
20% Don't Like
P (Other|Like video games)
P(Other Like Video Gam es 0.40 0.80
= 0.432
P ( Like Video Gam es) 0.60 0.70 0.40 0.80
6. The flow of cells through the switch board of your faculty is
distributed as Poisson with an average 120 cells per hour
1) What is the prob. of 3 calls per minute
2) What is the prob. of 5 calls per 2 minutes
3) What is the prob. of NO calls per 0.5 minute
4) What is the prob. of more than 2 calls within in 2 minutes
120
1) µ = = 2 calls/min
60
3 4
P(x=3)= 2 = 1.34
3 !
120
2) µ = X 2 = 4 calls/2min
60
5 4
P(x=5)= 4 = 0.156
5 !
120 1
3) µ = X = 1 calls/ 1 2 min
60 2
0 1
P(x=0)= 1 = 0.367
0 !
4) µ = 4 calls/ 2 minutes
P(x>2)= 1- P(x ≤ 2)
2 4 1 4 0 4
=1- 4 4 4 = 0.238
2 ! 1 ! 0 !
7. Assume the mean number of errors for a chapter in same
book is 0.8, what is the probability that there are less than two errors
in a particular chapter in this book?
µ = 0.8
P(x2) = 1- P(x<2) = 1-[P(x=0) +P(x=1) +P(x=2)]
= 1- [0.135 + 0.27 + 0.27] = 1- 0.675=0.325
c) P (x≥2) = 1-P(x<2) = 1-[P(x=0) + P(x=1)]
= 1- [0.135 + 0.27] = 1- 0.405 = 0.595
d) P (3